Group Element Commutes with Inverse

Theorem

Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Let $x \in G$.

Then:

$x \circ x^{-1} = x^{-1} \circ x$

That is, $x$ commutes with its inverse $x^{-1}$.

Proof

By definition of inverse element:

$x \circ x^{-1} = e = x^{-1} \circ x$

Hence the result by definition.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 4.6$: Example $86$